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Economic Price Optimization with Globally Linear Demand — Both Useful and Useless

February 2013 Pricing 1 Comment

Economic price optimization has a lot to tell us, but not necessarily the price which optimizes the firm’s profits.  Economic price optimization relies on a defined demand curve.  Unfortunately, defining the demand curve with sufficient precision and reliability for decision making can be done for only a subset of challenges.  So if this method is not useful for pricing in many cases, what use is it?

In this article, I will attempt to show mathephobic readers how to “read equations” to address the following:

  • What economic price optimization and globally linear demand function are;
  • The usefulness of this approach for setting prices;
  • The insights executives should gain from this approach.

Step 1:  The Firm’s Profit Equation

Economic price optimization refers to finding the price which will maximize the firm’s profits.  It does so by taking the first derivative of the firm’s profit equation with respect to price, setting this equal to zero, and finding the price which satisfies the resulting equation.  Let me demonstrate:

Start with the standard form of the firm’s profit equation Profit_Equation_of_the_Firm

where π stand for the firm’s profit, Q stands for the quantity sold, P stands for the price of the product, V stands for the variable costs to make the product, and F stands for the firm’s fixed costs.  There is nothing new in this equation.  It is taught in freshman business classes.

Step 2:  Set the Derivative with Respect to Price Equal to Zero

As freshman calculus tells us, the first derivative of profits with respect to price equals zero where the profit function has reached a maximum for normal products.  The price which delivers the maximum profits is clearly the optimal price from the firm’s perspective.

In looking at the profit equation of the firm, we can see that a price change affects the firm’s profit directly through the variable P. We can also expect that a price change influences the quantity sold and so indirectly affects the firm’s profit through the variable Q.  As for F and V, fixed costs and variable costs are constants with respect to a pure price change.

Taking the first derivative of the firm’s profit equation with respect to price and setting this equal to zero yields

Price_Optimization_Linear_Demand
Step 3:  Use the Demand Function

The derivative of the firm’s profit equation depends upon the relationship between price (P) and quantity sold (Q). The demand function defines this relationship.

At this point, most economists make a simplifying assumption of the shape of the demand function.  Given that higher prices are associated with lower sales, and lower prices deliver higher sales for normal goods, we know the demand function must slope downward.  A good simple function which slopes down is a straight line.  Hence, economists often assume a globally linear downward sloping curve when demonstrating economic price optimization in freshman economics classes.  (Given market research or sufficient data, the actual demand function can be used, but that is a completely different article.)

No decent economist actually believes demand curves are globally linear, but we all like them because they are simple and instructive.  Hence, whatever follows from here must be taken not from the viewpoint of “this will give us the definitive and quantitatively accurate best price”.  It will not.  But it can be taken from the viewpoint of “this might give us some insights into pricing and corporate strategy”.

A globally linear demand function is defined as that which the quantity demanded by the market varies linearly with the price extracted by the firm.  It would look like:

Linear_Demand_Curve

 

or be described mathematically as:

Linear_Demand_Function

 

Where QM is the maximum demand possible in the market (the quantity demanded when the price is zero, that is, free) and S is the maximum price any one unit can be sold.  Conceptually, S is a very powerful issue.  It is simultaneously the maximum utility one item can provide anyone in the market and the maximum willingness to pay within the market, while also reflecting the maximum benefits delivered.

Step 4:  Insert, Simplify, & Identify

Inserting the demand function into the above questions and simplifying reveals the following identities.

The optimal price is

Optimal_Price_Linear_Demand

The quantity sold at this price is

Optimal_Quantity_Linear_Demand

And the firm’s profit at this price is

Optimal_Profit_Linear_Demand

(Try getting these equations on your own … or better yet, have a high school senior taking calculus derive it.  It really is that simple.)

Step 5:  Interpret

Now again, these aren’t equations an executive can actually use to set prices, set production, or predict profits.  They were derived using an expression for the demand function which we know is a lie.  So what good is it?  Is it GIGO (Garbage In Garbage Out)?  Should we throw it away along with this whole approach?  In regard to actually setting prices, the answer is yes; with respect to strategic corporate  insights, emphatically no.

These equations may not tell us the optimal price nor predict profit, but they do tell us a lot about competitive strategy.

Starting with the equation for optimal price:  Notice that optimal price (P) increases with the maximum utility (S) and the variable costs (V).  Let’s focus on the S part.

The firm’s optimal price increases when the value of the product to its market increases—that is when S increases.  Hence, when a firm wants to charge higher prices, it should seek to enhance the benefits of their offerings and make their customers aware of these benefits.  This is precisely why product managers need to focus on identifying the goals of their customers and developing solutions to their needs.  It is also why sales executives are smart to practice value-based selling and marketing executives should spend money on advertisements to convince customers of the benefits of their products.

As for the V part:  executives often focus on cost reductions in order to ensure their prices are in line with the markets.  The above equations support this approach and I agree that firms need to manage costs to compete.  But this is only half of the story, as told by these equations.  Benefits are the other half– benefits delivered to customers.

Now, let’s turn to the firm’s profits at the optimal price.  Notice that profits are dependent on the difference between S and V, squared.  That is, the greater the difference between the utility of a product for a customer and the cost to produce the product at the firm, the more the firm makes.  And the profits aren’t just linearly dependent upon the difference between the benefits delivered and the costs to create, they are quadratically dependent, that is, for every doubling of difference there is a quadrupling of profit.

This relation between profit and the difference of S and V defines much of modern competitive strategy.

  • A firm is considered to have a competitive advantage if it can earn more profit than its competitors in the same market.
  • If a firm wants a competitive advantage over its competitors, it needs some strategic resource that its competitors do not and cannot have (rare and inimitable).
  • Moreover, that strategic resource will be strategic precisely because it enables the firm to deliver more benefits to its customers than its competitors without increasing costs, or it enables the firm to reduce costs without reducing benefits, or do both concurrently.

Put in relation to our equations, a strategic resource delivers a competitive advantage precisely because it increases the difference between S and V.  That is the difference between the maximum benefits delivered and the costs to produce, in comparison to its competitors.



  • Ken Nicholson

    Tim, thanks for taking the time to teach those of us fighting price in the trenches the concepts that can improve our strategic thinking. Well written as usual putting it straight and simple.

About the author

Tim J. Smith, PhD is the Managing Principal of Wiglaf Pricing, and an Adjunct Professor at DePaul University of Marketing and Economics. His most recent book is Pricing Strategy: Setting Price Levels, Managing Price Discounts, & Establishing Price Structures.

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